PARADISE LOST? ayeing the gate apprehensively under the stern regard of the uncompromising Dutchman. The postulational method for securing freedom from contradiction proposed by Hilbert will, says, Brouwer, accomplish its end - produce no contradictions, but 'nothing of mathematical value will be attained hi this manner; a false theory which is not stopped by a contradiction is none the less false, just as a criminal policy unchecked by a reprimanding court is none the less criminal.' The root of Brouwer's objection to the 'criminal policy' of his opponents is something new - at least in mathematics. He objects to an unrestricted use of Aristotelian logic, particularly in dealing with infinite sets, and he maintains that such logic is bound to produce contradictions when applied to sets which cannot be definitely constructed in Kronecker's sense (a rule of procedure must be given whereby the things in the set can be produced). The law of 'excluded middle' (a thing must have a certain property or must not have that property, as for example in the assertion that a number is prime or is not prime) is legitimately usable only when applied to finite sets. Aristotle devised his logic as a body of working rules for finite sets, basing his method on human experience of finite sets, and there is no reason whatever for supposing that a logic which is ade- quate for the finite will continue to produce consistent (not contradictory) results when applied to the infinite. This seems reasonable enough when we recall that the very definition of an infinite set emphasizes that a part of an infinite set may contain precisely as many things as the whole set (as we have illustrated many times), a situation which never happens for a finite set when 'part* means some, but not all (as it does in the definition of an infinite set). Here we have what some consider the root of the trouble in Cantor's theory of the actual infinite. For the definition of a set (as stated some time back), by which all things having a certain quality are 'united' to form a 'set' (or 'class'), is not suitable as a basis for the theory of sets, in that the definition either is not constructive (in Kronecker's sense) or assumes a constructibility which no mortal can produce. Brouwer claims that the use of the law of excluded middle in such a situation is at best merely 637